The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.8.12. Let $R$ be a ring and let $M$ be an $R$-module. Then $M$ is the colimit of a directed system $(M_ i, \mu _{ij})$ of $R$-modules with all $M_ i$ finitely presented $R$-modules.

Proof. Consider any finite subset $S \subset M$ and any finite collection of relations $E$ among the elements of $S$. So each $s \in S$ corresponds to $x_ s \in M$ and each $e \in E$ consists of a vector of elements $f_{e, s} \in R$ such that $\sum f_{e, s} x_ s = 0$. Let $M_{S, E}$ be the cokernel of the map

\[ R^{\# E} \longrightarrow R^{\# S}, \quad (g_ e)_{e\in E} \longmapsto (\sum g_ e f_{e, s})_{s\in S}. \]

There are canonical maps $M_{S, E} \to M$. If $S \subset S'$ and if the elements of $E$ correspond, via this map, to relations in $E'$, then there is an obvious map $M_{S, E} \to M_{S', E'}$ commuting with the maps to $M$. Let $I$ be the set of pairs $(S, E)$ with ordering by inclusion as above. It is clear that the colimit of this directed system is $M$. $\square$


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